The present invention relates to optical transmission techniques. Optical transmission in a fiber is given by the nonlinear Schrödinger equation (NLSE).
                                          ∂            E                                ∂            z                          =                              (                                          N                ^                            +                              D                ^                                      )                    ⁢          E                                    (        1        )            where
            D      ^        =                            -                      1            2                          ⁢        α            -                        β          1                ⁢                  ∂                      ∂            t                              -              j        ⁢                                  ⁢                  β          2                ⁢                  1                      2            !                          ⁢                              ∂            2                                ∂                          t              2                                          +                        β          3                ⁢                  1                      3            !                          ⁢                              ∂            3                                ∂                          t              3                                      ⁢                                  ⁢        and                        N      ^        =          j      ⁢                          ⁢              γ        ⁡                  [                                                                                        E                                                  2                            ⁢              I                        -                                          1                3                            ⁢                              (                                                      E                    H                                    ⁢                                      σ                    3                                    ⁢                  E                                )                            ⁢                              σ                3                                              ]                    are the linear and nonlinear operators. In the linear operator, α, β1, β2, β3 are 2×2 matrices representing attenuation, polarization-mode dispersion, group velocity dispersion and dispersion slope; whereas in the nonlinear operator, γ is the fiber's nonlinear parameter.
The noise sources in fiber optic transmission include amplified spontaneous emission (ASE) of inline erbium-doped fiber amplifiers (EDFA), and shot noise and thermal noise of the receiver. For a linear channel, the capacity per bandwidth is given by Shannon's limit: C=log2(1+η), where η is the signal-to-noise ratio (SNR). In the absence of nonlinearity, it is possible to increase capacity by boosting the signal power in order to increase SNR. In optical fiber, however, the variance of signal distortion arising from the Kerr nonlinearity grows faster than SNR. It has been shown that fiber nonlinearity limits the usable signal power, and hence the achievable capacity-distance product. The statistical interaction between signal and in-band noise through fiber nonlinearity ultimately imposes a “nonlinear Shannon's limit” on the achievable capacity. However, nonlinear signal-signal interactions are deterministic, and can be mitigated via nonlinear compensation. Reduction of such deterministic signal distortion at the receiver can improve the capacity-distance product.
In the absence of noise, the signal propagation equation in (1) can be
                                          ∂            E                                ∂            z                          =                              -                          (                                                N                  ^                                +                                  ξ                  ⁢                                                                          ⁢                                      D                    ^                                                              )                                ⁢          E                                    (        2        )            This operation is analogous to passing the received signal through a fictitious link where each element in the fictitious link exactly inverts the elements of the forward-propagating channel. In the presence of noise however, the inverse NLSE is inexact, leading to irreducible signal distortion from which the nonlinear Shannon's limit arise. All nonlinear compensation (NLC) methods are ultimately based on approximating the inverse NLSE. Depending on the dispersion map, the degree of nonlinearity, the signal modulation format and the spectral characteristics of the signal, different algorithms can be implemented to approximate. Thus, NLC algorithms trade-off between algorithmic-complexity and achievable performance.
Some well-known NLC methods include:
a) Nonlinear signal de-rotation, in which the transmission link is assumed to be a lumped nonlinearity followed by pure dispersion. This technique has been demonstrated in simulations and offline experiments for both single-carrier (SC) and orthogonal frequency-division multiplexed (OFDM) signaling; and its performance depends on the dispersion map being well-managed to keep accumulated dispersion within a symbol where signal power is appreciable.
b) Recently, digital back-propagation (DBP) using the split-step Fourier method (SSFM) has been demonstrated. The SSFM itself is a well-known technique that has been developed for forward propagation, and works by dividing a fiber link into sufficiently small steps such that at the end of each step, the phase rotation in time and frequency due to {circumflex over (N)} or {circumflex over (D)} is small enough to preserve the accuracy of the final solution.
Unlike nonlinear signal de-rotation, DBP is a universal NLC method and is independent of the modulation format and system dispersion map. DBP can be implemented at the transmitter, at the receiver, or a combination of both transmitter and receiver. In receiver-side DBP, an optical-to-electrical down-converter recovers the electric field in one or both polarizations of the fiber. The electronic signal is sampled with a high-speed analog-to-digital converter (ADC), where a digital signal processor (DSP) computes the inverse NLSE of the digitized waveform. In transmitter-side DBP, the inverse NLSE is calculated by a DSP at the transmitter. The pre-distorted signal is then used to drive optical modulators via arbitrary-waveform generators (AWG).